Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized for You

we will pick new questions that match your level based on your Timer History

Track Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

Equilateral triangle BDF is inscribed in equilateral triangl [#permalink]

Show Tags

19 Jun 2010, 08:17

2

This post received KUDOS

6

This post was BOOKMARKED

00:00

A

B

C

D

E

Difficulty:

95% (hard)

Question Stats:

43% (02:33) correct
57% (01:55) wrong based on 217 sessions

HideShow timer Statistics

Attachment:

Captura.jpg [ 108.93 KiB | Viewed 7502 times ]

Equilateral triangle BDF is inscribed in equilateral triangle ACE, as shown in the figure above. The shaded region is what fraction of the area of the triangle ACE?

(1) Angle DFE is 90º (2) The length of AF is 10*sqrt(3)

Re: DS: Equilateral triangle inscribed in triangle [#permalink]

Show Tags

19 Jun 2010, 09:09

3

This post received KUDOS

Expert's post

3

This post was BOOKMARKED

Nusa84 wrote:

Equilateral triangle BDF is inscribed in equilateral triangle ACE, as shown in the figure above. The shaded region is what fraction of the area of the triangle ACE? 1. Angle DFE is 90º 2. The length of AF is 10*sqrt(3)

Can you help me with this one? Thanks!!!

The problem is easier to solve than to explain, but anyway:

In equilateral triangle all angles equal to 60 degrees and \(Area_{equilateral}=\frac{a^2\sqrt{3}}{4}\), where \(a\) is the length of a side.

(1) Angle DFE is 90º --> angles ABF and BDC must also be 90º (for example \(\angle {BDC}=180-\angle {BDF}-\angle{EDF}=180-60-30\) and the same for ABF). Also as \(\angle{DEF}=\angle{BCD}=\angle{BAF}=60\), then triangles DFE, BCD and BAF are 30-60-90 trianlges. In such triangle sides are in the ratio: \(1:\sqrt{3}:2\) (smallest side (1) is opposite the smallest angle (30°), and the longest side (2) is opposite the largest angle (90°)).

So if \(DE=2x\) (hypotenuse in right triangle DFE), then \(DC=x\) (smaller leg in right triangle BCD) and \(BD=\sqrt{3}x\) (larger leg in right triangle BCD, also the side of inscribed triangle). So the side of triangle ACE would be \(CE=DC+DE=x+2x=3x\)

Area of the shaded region (right triangle BDC) would be \(Area_{BDC}=\frac{BD*DC}{2}=\frac{\sqrt{3}x*x}{2}=\frac{\sqrt{3}x^2}{2}\) and the area of equilateral triangle \(Area_{ACE}=\frac{a^2\sqrt{3}}{4}=\frac{(3x)^2*\sqrt{3}}{4}=\frac{9x^2\sqrt{3}}{4}\);

Re: DS: Equilateral triangle inscribed in triangle [#permalink]

Show Tags

19 Jun 2010, 09:27

Bunuel wrote:

Nusa84 wrote:

Equilateral triangle BDF is inscribed in equilateral triangle ACE, as shown in the figure above. The shaded region is what fraction of the area of the triangle ACE? 1. Angle DFE is 90º 2. The length of AF is 10*sqrt(3)

Can you help me with this one? Thanks!!!

Attachment:

Captura.jpg

The problem is easier to solve than to explain, but anyway:

In equilateral triangle all angles equal to 60 degrees and \(Area_{equilateral}=\frac{a^2\sqrt{3}}{4}\), where \(a\) is the length of a side.

(1) Angle DFE is 90º --> angles ABF and BDC must also be 90º (for example \(\angle {BDC}=180-\angle {BCD}-\angle{EDF}=180-60-30\) and the same for ABF). Also as \(\angle{DEF}=\angle{BCD}=\angle{BAF}=60\), then triangles DFE, BCD and BAF are 30-60-90 trianlges. In such triangle sides are in the ratio: \(1:\sqrt{3}:2\) (smallest side (1) is opposite the smallest angle (30°), and the longest side (2) is opposite the largest angle (90°)).

So if \(DE=2x\) (hypotenuse in right triangle DFE), then \(DC=x\) (smaller leg in right triangle BCD) and \(BD=\sqrt{3}x\) (larger leg in right triangle BCD, also the side of inscribed triangle). So the side of triangle ACE would be \(CE=DC+DE=x+2x=3x\)

Area of the shaded region (right triangle BDC) would be \(Area_{BDC}=\frac{BD*DC}{2}=\frac{\sqrt{3}x*x}{2}=\frac{\sqrt{3}x^2}{2}\) and the area of equilateral triangle \(Area_{ACE}=\frac{a^2\sqrt{3}}{4}=\frac{(3x)^2*\sqrt{3}}{4}=\frac{9x^2\sqrt{3}}{4}\);

Re: DS: Equilateral triangle inscribed in triangle [#permalink]

Show Tags

06 Mar 2014, 19:12

Bunuel wrote:

Nusa84 wrote:

Equilateral triangle BDF is inscribed in equilateral triangle ACE, as shown in the figure above. The shaded region is what fraction of the area of the triangle ACE? 1. Angle DFE is 90º 2. The length of AF is 10*sqrt(3)

Can you help me with this one? Thanks!!!

Attachment:

Captura.jpg

The problem is easier to solve than to explain, but anyway:

In equilateral triangle all angles equal to 60 degrees and \(Area_{equilateral}=\frac{a^2\sqrt{3}}{4}\), where \(a\) is the length of a side.

(1) Angle DFE is 90º --> angles ABF and BDC must also be 90º (for example \(\angle {BDC}=180-\angle {BCD}-\angle{EDF}=180-60-30\) and the same for ABF). Also as \(\angle{DEF}=\angle{BCD}=\angle{BAF}=60\), then triangles DFE, BCD and BAF are 30-60-90 trianlges. In such triangle sides are in the ratio: \(1:\sqrt{3}:2\) (smallest side (1) is opposite the smallest angle (30°), and the longest side (2) is opposite the largest angle (90°)).

So if \(DE=2x\) (hypotenuse in right triangle DFE), then \(DC=x\) (smaller leg in right triangle BCD) and \(BD=\sqrt{3}x\) (larger leg in right triangle BCD, also the side of inscribed triangle). So the side of triangle ACE would be \(CE=DC+DE=x+2x=3x\)

Area of the shaded region (right triangle BDC) would be \(Area_{BDC}=\frac{BD*DC}{2}=\frac{\sqrt{3}x*x}{2}=\frac{\sqrt{3}x^2}{2}\) and the area of equilateral triangle \(Area_{ACE}=\frac{a^2\sqrt{3}}{4}=\frac{(3x)^2*\sqrt{3}}{4}=\frac{9x^2\sqrt{3}}{4}\);

Re: Equilateral triangle BDF is inscribed in equilateral triangl [#permalink]

Show Tags

06 Mar 2014, 23:55

Expert's post

Nusa84 wrote:

Attachment:

The attachment Captura.jpg is no longer available

Equilateral triangle BDF is inscribed in equilateral triangle ACE, as shown in the figure above. The shaded region is what fraction of the area of the triangle ACE?

(1) Angle DFE is 90º (2) The length of AF is 10*sqrt(3)

Note that in a given equilateral triangle, you can inscribe many other equilateral triangles. The areas of these inscribed triangles will be different hence the leftover areas will be different.

Attachment:

Ques3.jpg [ 13.11 KiB | Viewed 4420 times ]

So depending on what type of triangle is inscribed, the shaded region will be a fraction of the area of triangle ACE.

(1) Angle DFE is 90º

All internal angles of Equilateral triangles are 60. Angle DFE is 90 so triangle DFE is a right triangle with angles 30-60-90 and sides in the ratio 1:√3:2.

Attachment:

Ques4.jpg [ 15.69 KiB | Viewed 4415 times ]

If we assume that side of inscribed triangle is s, we get the side of the large triangle in terms of s too. We also get the area of the right triangle (shaded region) in terms of s. Hence we can easily get the area of the shaded region as a fraction of area of triangle ACE (since they will both be in terms of s^2). This statement alone will be sufficient.

(2) The length of AF is 10*sqrt(3) Look at the three triangles shown above again. AF could be 10√3 in any of the three cases. The area of shaded region as a fraction of the area of ACE will be different in each case. Hence this statement alone is not sufficient to answer the question.

Re: Equilateral triangle BDF is inscribed in equilateral triangl [#permalink]

Show Tags

07 Mar 2014, 00:05

Since you asked for further discussion, here goes.

I more intuited the answer than solved. I know that each interior angle of both equilateral triangles is 60. If I am told that DFE = 90, I can figure out that the other 3 triangles are 30-60-90 and congruent (Each 60-degree angle is across from one side of the smaller equilateral triangle; thus all corresponding sides for the right triangles are the same "x(sqrt3)"). Because I know the relative dimensions of triangles BCD, ABF, DFE, and BDF, I can figure out the relative areas of the triangles. For example, I know I can make BD sqrt3, BC 2, and CD 1. I can figure out the areas of all 4 smaller triangles, add these areas to get ACE's area, and thus determine the fraction of ACE that BCD represents. I didn't actually bother to assign values or do the calculations because it was clear Statement 1 is sufficient.

Eliminate B, C, and E.

Statement 2 only tells me AF. It doesn't provide me with any information about any other sides or angles in the figure. Thus, I would not be able to solve for area. The information is useless. Insufficient.

Re: DS: Equilateral triangle inscribed in triangle [#permalink]

Show Tags

07 Mar 2014, 01:34

Expert's post

X017in wrote:

Bunuel wrote:

Nusa84 wrote:

Equilateral triangle BDF is inscribed in equilateral triangle ACE, as shown in the figure above. The shaded region is what fraction of the area of the triangle ACE? 1. Angle DFE is 90º 2. The length of AF is 10*sqrt(3)

Can you help me with this one? Thanks!!!

Attachment:

Captura.jpg

The problem is easier to solve than to explain, but anyway:

In equilateral triangle all angles equal to 60 degrees and \(Area_{equilateral}=\frac{a^2\sqrt{3}}{4}\), where \(a\) is the length of a side.

(1) Angle DFE is 90º --> angles ABF and BDC must also be 90º (for example \(\angle {BDC}=180-\angle {BCD}-\angle{EDF}=180-60-30\) and the same for ABF). Also as \(\angle{DEF}=\angle{BCD}=\angle{BAF}=60\), then triangles DFE, BCD and BAF are 30-60-90 trianlges. In such triangle sides are in the ratio: \(1:\sqrt{3}:2\) (smallest side (1) is opposite the smallest angle (30°), and the longest side (2) is opposite the largest angle (90°)).

So if \(DE=2x\) (hypotenuse in right triangle DFE), then \(DC=x\) (smaller leg in right triangle BCD) and \(BD=\sqrt{3}x\) (larger leg in right triangle BCD, also the side of inscribed triangle). So the side of triangle ACE would be \(CE=DC+DE=x+2x=3x\)

Area of the shaded region (right triangle BDC) would be \(Area_{BDC}=\frac{BD*DC}{2}=\frac{\sqrt{3}x*x}{2}=\frac{\sqrt{3}x^2}{2}\) and the area of equilateral triangle \(Area_{ACE}=\frac{a^2\sqrt{3}}{4}=\frac{(3x)^2*\sqrt{3}}{4}=\frac{9x^2\sqrt{3}}{4}\);

Equilateral triangle BDF is inscribed in a equilateral triangle ACE, [#permalink]

Show Tags

04 Apr 2015, 09:29

Equilateral triangle BDF is inscribed in a equilateral triangle ACE, as shown in the figure above. The shaded region is what fraction of the area of triangle ACE?

1) Angle DFE = 90º

2) The length of AF is 10\sqrt{3}

Attachments

quant_ds_00000225.jpg [ 19.57 KiB | Viewed 2705 times ]

Re: Equilateral triangle BDF is inscribed in a equilateral triangle ACE, [#permalink]

Show Tags

04 Apr 2015, 10:16

Awli wrote:

Equilateral triangle BDF is inscribed in a equilateral triangle ACE, as shown in the figure above. The shaded region is what fraction of the area of triangle ACE?

Only difference that I find in second statement: 10*sqrt(3) in original task and 10\sqrt{3} in this task But this difference change nothing.

1) If we know that DFE is right triangle 30:60:90, so we know ratio of its sides, can find side of big triangle (relative to DFE), calculate their areas and find fraction

2) This statement doesn't give us enough information.

Re: Equilateral triangle BDF is inscribed in equilateral triangl [#permalink]

Show Tags

05 Apr 2015, 05:01

Expert's post

Awli wrote:

Equilateral triangle BDF is inscribed in a equilateral triangle ACE, as shown in the figure above. The shaded region is what fraction of the area of triangle ACE?

1) Angle DFE = 90º

2) The length of AF is 10\sqrt{3}

Merging topics. Pleas search before posting. Thank you. _________________

Re: Equilateral triangle BDF is inscribed in equilateral triangl [#permalink]

Show Tags

17 Apr 2016, 06:59

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________

Re: Equilateral triangle BDF is inscribed in equilateral triangl [#permalink]

Show Tags

03 May 2016, 00:03

VeritasPrepKarishma wrote:

Nusa84 wrote:

Attachment:

Captura.jpg

Equilateral triangle BDF is inscribed in equilateral triangle ACE, as shown in the figure above. The shaded region is what fraction of the area of the triangle ACE?

(1) Angle DFE is 90º (2) The length of AF is 10*sqrt(3)

Note that in a given equilateral triangle, you can inscribe many other equilateral triangles. The areas of these inscribed triangles will be different hence the leftover areas will be different.

Attachment:

Ques3.jpg

So depending on what type of triangle is inscribed, the shaded region will be a fraction of the area of triangle ACE.

(1) Angle DFE is 90º

All internal angles of Equilateral triangles are 60. Angle DFE is 90 so triangle DFE is a right triangle with angles 30-60-90 and sides in the ratio 1:√3:2.

Attachment:

Ques4.jpg

If we assume that side of inscribed triangle is s, we get the side of the large triangle in terms of s too. We also get the area of the right triangle (shaded region) in terms of s. Hence we can easily get the area of the shaded region as a fraction of area of triangle ACE (since they will both be in terms of s^2). This statement alone will be sufficient.

(2) The length of AF is 10*sqrt(3) Look at the three triangles shown above again. AF could be 10√3 in any of the three cases. The area of shaded region as a fraction of the area of ACE will be different in each case. Hence this statement alone is not sufficient to answer the question.

Answer (A)

thanks , quite a good explanation. Even easier if we put FE as "s" and now internal Equilateral trianle has sq root(3)* s as side. Nw straight to shaded region . we can get area and fraction by same 30-6-90.

gmatclubot

Re: Equilateral triangle BDF is inscribed in equilateral triangl
[#permalink]
03 May 2016, 00:03

MBA Admission Calculator Officially Launched After 2 years of effort and over 1,000 hours of work, I have finally launched my MBA Admission Calculator . The calculator uses the...

Final decisions are in: Berkeley: Denied with interview Tepper: Waitlisted with interview Rotman: Admitted with scholarship (withdrawn) Random French School: Admitted to MSc in Management with scholarship (...

The London Business School Admits Weekend officially kicked off on Saturday morning with registrations and breakfast. We received a carry bag, name tags, schedules and an MBA2018 tee at...